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VECTOR ADDITION

VECTOR ADDITION. BY. SMA N 1 SLAWI. FI KIRANKU. KA MU. S ELALU. I NGAT. VECTORS ADDITION. 3 kg + 4 kg =. 7 kg. ?. 3 N + 4 N =. Vectors Quantities and Scalar Quantities. Vector Quantities are physical quantities which have a magnitude or value and direction

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VECTOR ADDITION

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  1. VECTOR ADDITION BY SMA N 1 SLAWI

  2. FIKIRANKU KAMU SELALU INGAT VECTORS ADDITION

  3. 3 kg + 4 kg = 7 kg ? 3 N + 4 N =

  4. Vectors Quantities and Scalar Quantities • Vector Quantities are physical quantities which have a magnitude or value and direction Example : velocity, acceleration, Force, etc • Scalar quantities are physical quantities which have magnitude or value without direction Example : Mass, Time, Temperature, volume , etc

  5. Notation Vectors The vectors quantities are written in bold type, while italicization is used to represent the scalar value/scalar quantities. Exp: -The vector A is written as A and the scalar quantity is written A The vector quantities can written with a distinguishing mark, such as an arrow. Exp: - The vector A is written as A and the scalar quantity is written A .

  6. A Vector can be expressed in diagram with a directed line segment. Direction of vector Capture point A Magnitude of vector  = direction of vector 

  7. NEGATIVE VECTORS Negative vectors is the vectors which have the same in magnitude but opposite in direction A - A

  8. To determine resultant vector with graphic method 1. Polygon method 2. Parallelogram method

  9. F1 BOX F1 + F2 F2 F1 F2 TRIANGLE METHOD F3 F2 F3 F1 + F2 + F3 F1 BOX F1 POLYGON METHOD F2

  10. Parallelogram Method FR = F1 + F2 F1 F1 BOX F2 F2

  11. Characteristic of Vectors addition

  12. Resultant vector with analytical method • Cosine equation • Vector component method

  13. The Magnitude and direction of vector resultant with cosines equation FR = F1 + F2 F1 F1  1   2 F2 F2 The Magnitude of vector resultant : The direction of vector resultant :

  14. Two vectors form angle of 00 F1 F2 F1 F2 FR = F1 + F2 The magnitude resultant of vector The direction of vector resultant a direction with both of vectors

  15. Two vectors form angle of 1800(Two Vectors opposite each other) F1 F2 F1 FR = F1 - F2 F2 The magnitude resultant of vector The direction of vector resultant a direction with the biggest vector

  16. Two vectors form angle of 900 (Two Vectors perpendicular each other) The magnitude resultant of vector FR = F1 + F2 F2 The direction of vector resultant 900  F1

  17. 2. vectors resultant with component vectors method Vectors components : Fx = F cos Fy = F sin The Magnitude of vectors resultant F: For two or more vectors : y F Fy  X Fx The Direction of vector resultant :

  18. SAMPLE PROBLEM • A vector of velocity (V) forms an angle 300 with positive X axis andthe magnitude is 20 m/s. determine the magnitude of vector component! • Two vector of velocity have base points which coincide, those are v1= 3 m/s and v2 = 4 m/s. if  = 600.find the magnitude and direction of vector resultant. • Four velocity vector have magnitudes and directions as follows : V1 = 10 m/s , 1 = 00 V2 = 12 m/s , 2 = 600 V3 = 10 m/s , 3 = 1200 V4 = 6 m/s , 4 = 2400 Determine the magnitude and direction of vector resultant!

  19. 1. SOLUTION The components of vector V Vy  Vx

  20. 2. v1 v   V2

  21. v1x = v1cos1 v1y = v1 sin 1 = 10 cos00 = 10 sin 00 = 10 (1) = 10 (0) = 10 = 0 v2x = v2cos2 v2y = v2 sin 2 = 12 cos 600 = 12 sin 600 = 12 ( ) = 12 ( ) = 6 = 6 v3x = v3cos3 v3y = v3 sin 3 = 10 cos 1200 = 10 sin 1200 = 10 ( ) = 10 ( ) = 5 = - 5 v4x = v4cos4 v4y = v4 sin 4 = 6 cos 2400 = 6 sin 2400 = 6 ( ) = 6 ( ) = -3 = -3

  22. Table The magnitude of result vector The direction of result vector

  23. V2y V2 V3 V3y 600 600 V1 V4x V3x 600 V2x V4y V4 V2x = V2 cos 600 V2y = V2 sin 600 V3x = V3 cos 600 V3y = V3 sin 600 V4x = V4 cos 600 V4y = V4 sin 600

  24. V2y V2 V3 V3y 300 300 V1 V4x V3x V2x 300 V4y V4 V2x = V2 sin 300 V2y = V2 cos 300 V3x = V3 sin 300 V3y = V3 cos 300 V4x = V4 sin 300 V4y = V4 cos 300

  25. V2y V2 V3 V3y 300 600 V1 V4x V3x 600 V2x V4y V4 V2x = V2 cos 600 V2y = V2 cos 600 V3x = V3 sin 600 V3y = V3 cos 600 V4x = V4 cos 600 V4y = V4 cos 600

  26. UNIT VECTOR • Unit vector is a vector of which the magnitude equals to one and the direction is the same as the direction of vector component. • In three dimensional case there are 3 umit vector, that is i , j , k • i = unit vector in the same direction as x axis • j = unit vector in the same direction as y axis • k = unit vector in the same direction as z axis

  27. Unit vector in three dimensional case Z k j Y i X

  28. Vector A can be expressed by unit vector as follows A = AXi +Ay j +Az k Z The magnitude of vector A can be expressed by A Y AX i Az k Ay j X In one dimensional case, then Ay = Az = 0 In two dimensional case , then Az = 0

  29. Vector Multiplication • Dot Product Vector Dot Product vector gives a scalar result, therefore the dot product vector is also called scalar product vector. The dot product vector between A and B can be expressed as follows : A.B = A B cos A = vector A, B = vector B, A = the magnitude of vector A B = the magnitude of vector B,  = angle between A and B

  30. A = AXi +Ayj +Azk B = BXi +Byj +Bzk Dot product vector Characteristic a peer the unit vector i . i = j . j = k . k = (1) (1) cos 0 = 1 i . j = i . k = j . k = (1) (1) cos 900 = 0 j . i = k . i = k . j = (1) (1) cos 900 = 0 If vector A and vector B written in unit vector notation : and So, dot product vector A and vector B is A . B = (AXi +Ayj +Azk ) (BXi +Byj +Bzk ) = AXi BXi + AXi Byj +AXi Bz k + Ayj BXi + Ayj Byj + Ayj Bzk +Azk BXi +Azk Byj +Azk Bzk A . B = AX BX+Ay By +Az Bz

  31. Cross Product Vector Cross Product vector gives a new vector result, therefore the dot product vector is also called vector product. The Cross product vector between A and B can be product vector C, Which the magnitude is C = AXB = A B sin  A = vector A, B = vector B, A = the magnitude of vector A B = the magnitude of vector B,  = angle between A and B

  32. A = AXi +Ayj +Azk B = BXi +Byj +Bzk Cross product vector Characteristic a peer the unit vector i x i = j x j = k x k = (1) (1) sin 0 = 0 i x j = kj x i = -k j x k = i k x j = -i k x i = j i x k = -j If vector A and vector B written in unit vector notation : and So, cross product vector A and vector B is A X B = (AXi +Ayj +Azk ) (BXi +Byj +Bzk ) = AXi BXi + AXi Byj +AXi Bz k + Ayj BXi + Ayj Byj + Ayj Bzk +Azk BXi +Azk Byj +Azk Bzk

  33. = AXi Byj +AXi Bz k + Ayj BXi + Ayj Bzk +Azk BXi +Azk Byj = AX Byk+AX Bz (-j)+ Ay BX (-k) + Ay Bzi+Az BXj+Az By(-i) = Ay Bzi+Az By(-i) + Az BXj+ AX Bz (-j)+ AX Byk+ Ay BX (-k) = Ay Bzi-Az By(i) + Az BXj- AX Bz (j)+ AX Byk- Ay BX (k) = Ay Bzi-Az Byi+ Az BXj- AX Bz j+ AX Byk- Ay BXk A X B = (Ay Bz-Az By)i+(Az BX- AX Bz)j+ (AX By- Ay BX)k

  34. Cross product vector with determinant method A = AXi +Ayj +Azk B = BXi +Byj +Bzk - negative C = A x B i j k i j Ax Ay Az Ax Ay Bx By Bz Bx By C = + positive C= A X B = Ay Bzi-Az Byi+ Az BXj- AX Bz j+ AX Byk- Ay BXk C = A X B = (Ay Bz-Az By)i+(Az BX- AX Bz)j+ (AX By- Ay BX)k

  35. Cross product two vectors k j i + Positive - Negative

  36. SAMPLE PROBLEM A = 2i +3j +k A = AXi +Ayj +Azk B = BXi +Byj +Bzk B = 4i +2j -2k Determine : a A . B b. A x B SOLUTION 1. a A . B = AX BX+Ay By+Az Bz = (2) (4)+(3) (2)+(1)(-2) = 8 +6 - 2 = 12

  37. b. C = A X B = (Ay Bz-Az By)i+(Az BX- AX Bz)j+ (AX By- Ay BX)k = ( (3) (-2) –(1) (2))i+( (1)(4)–(2)(-2) )j+ ( (2) (2)–(3) (4) )k = - 8 i+ 8 j - 8 k

  38. Cross product vector with determinant method A = AXi +Ayj +Azk A = 2i +3j +k B = BXi +Byj +Bzk B = 4i +2j -2k negative - -12 k -2 i C = A x B 4 j i jk ij 231 2 3 4 2 -2 4 2 C = 4 k 4 j -6 i + positive C = A X B = -8 i+ 8 j- 8 k

  39. NOTES Sin, cos, tan table

  40. Kuadran IV ( 3600 -  ) Kuadran II (1800 -  ) Kuadran III ( 1800 +  )

  41. Example Cos 1200 =………. 1200 = di kudran II ( hanya sin yang positive) Cos 1200 = cos (180 -  ) = cos (1800 – 600 ) = - cos 600 = sin 2400 =………. 2400 = di kudran III ( hanya tan yang positive) sin 2400 = sin (180 +  ) = sin (1800 + 600 ) = - sin 600 =

  42. Letak kuadran sudut sebuah vektor

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